if $ a \cdot b= a \cdot c $ and $ a \times b = a \times c $, does that follow that $ b=c $?

Question

if $ a \cdot b= a \cdot c $ and $ a \times b = a \times c $, does that follow that $ b=c $?

I know that $ a \cdot b = a \cdot c $ doesn't guarantee $ b = c$ by proving by counter example(a = (1,0), b = (0, 1), c= (0, -1)).

I also know that $ a \times b = a \times c $ by setting a as (1, 0, 0), b as (0, 0, 1) and c as (1, 0, 1).

But if $ a \cdot b= a \cdot c $ and $ a \times b = a \times c $, does that follow that $ b=c $?

And I also want to know if there is any way that I can prove that $ a \times b = a \times c $ doesn't refer to $ b = c$ by some other means other than stating counter example?

Answer 1

$\def\\#1{{\bf#1}}$If $\\a,\\b,\\c$ are vectors in $\Bbb R^3$ and $\\a\ne\\0$, then this is true. Proof. Since $$\\a\cdot(\\b-\\c)=0\quad\hbox{and}\quad \\a\times(\\b-\\c)=\\0\ ,$$ the vector $\\b-\\c$ is both perpendicular and parallel to $\\a$. This is only possible if $\\b-\\c=\\0$.

Answer 2

Here's an algebraic tool: if $v$ and $w$ are vectors in three-dimensional space with $v \times w = 0$ and $v \neq 0$, then there is a scalar $r$ such that $w = rv$.